1 Foraging Model
- 1.1 Dive profiles per indidivuals
- 1.2 Data Statistics before track cut
2 Data Cleaning Rules.
- 2.1 Data statistics after track cut
- 2.2 Chains
3 Temporal Variation in Dive Behavior
4 Raw Argos Data
5 Comparison with 2d movement model
- 5.1 Chains
- 5.2 Posterior Checks

1 Foraging Model

The proportion of time an animal is in a feeding behavioral state.

Process Model

\[Y_{i,t+1} \sim Multivariate Normal(d_{i,t},σ)\]

\[d_{i,t}= Y_{i,t} + γ_{s_{i,g,t}}*T_{i,g,t}*( Y_{i,g,t}- Y_{i,g,t-1} )\]

\[ \begin{matrix} \alpha_{i,1,1} & \beta_{i,1,1} & 1-(\alpha_{i,1,1} + \beta_{i,1,1}) \\ \alpha_{i,2,1} & \beta_{i,2,1} & 1-(\alpha_{i,2,1} + \beta_{i,2,1}) \\ \alpha_{i,3,1} & \beta_{i,3,1} & 1-(\alpha_{i,3,1} + \beta_{i,3,1}) \\ \end{matrix} \] \[logit(\phi_{Behavior}) = \alpha_{Behavior_{t-1}} \] The behavior at time t of individual i on track g is a discrete draw. \[S_{i,g,t} \sim Cat(\phi_{traveling},\phi_{foraging},\phi_{resting})\]

Dive information is a mixture model based on behavior (S)

\(\text{Average dive depth}(\psi)\) \[ DiveDepth \sim Normal(dive_{\mu_S},dive_{\tau_S})\]

1.1 Dive profiles per indidivuals

Dive Profiles with Argos timestamps

1.2 Data Statistics before track cut

## # A tibble: 11 x 4
##    Animal     n argos  dive
##     <int> <int> <int> <int>
##  1 131111   548   173   375
##  2 131115  1029   179   850
##  3 131116  1932   457  1475
##  4 131127  9275  2495  6780
##  5 131128   112    52    60
##  6 131130  1165   151  1014
##  7 131132  2292   589  1703
##  8 131133  8497  2078  6419
##  9 131134  2098   653  1445
## 10 131136  6844  1970  4874
## 11 154187  1866   486  1380

## [1] 35658

2 Data Cleaning Rules.

All tracks start with an observed argos location.
A single track must have one argos location per time step.

2.1 Data statistics after track cut

## # A tibble: 11 x 2
##    Animal     n
##     <int> <int>
##  1 131111   454
##  2 131115  1020
##  3 131116  1920
##  4 131127  9084
##  5 131128   106
##  6 131130   513
##  7 131132  1733
##  8 131133  8195
##  9 131134  2043
## 10 131136  6724
## 11 154187  1847

## # A tibble: 11 x 2
##    Animal Tracks
##     <int>  <int>
##  1 131111      1
##  2 131115      1
##  3 131116      2
##  4 131127     11
##  5 131128      1
##  6 131130      1
##  7 131132      1
##  8 131133      4
##  9 131134      1
## 10 131136      3
## 11 154187      1

sink(“Bayesian/NestedDive.jags”) cat(" model{

pi <- 3.141592653589

#for each if 6 argos class observation error

for(x in 1:6){

##argos observation error##
argos_prec[x,1:2,1:2] <- argos_cov[x,,]

#Constructing the covariance matrix
argos_cov[x,1,1] <- argos_sigma[x]
argos_cov[x,1,2] <- 0
argos_cov[x,2,1] <- 0
argos_cov[x,2,2] <- argos_alpha[x]
}

for(i in 1:ind){
for(g in 1:tracks[i]){

## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec[1,1:2,1:2])

#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)

###First Behavioral State###
state[i,g,1] ~ dcat(lambda[]) ## assign state for first obs

#Process Model for movement
for(t in 2:(steps[i,g]-1)){

#Behavioral State at time T
phi[i,g,t,1] <- alpha[state[i,g,t-1]] 
phi[i,g,t,2] <- 1-phi[i,g,t,1]
state[i,g,t] ~ dcat(phi[i,g,t,])

#Substate
sub_phi[i,g,t,1] <- sub_alpha[state[i,g,t]] 
sub_phi[i,g,t,2] <- 1-sub_phi[i,g,t,1]
sub_state[i,g,t] ~ dcat(sub_phi[i,g,t,])

#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])

#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])

#Gaussian Displacement in location
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)


}

#Final behavior state
phi[i,g,steps[i,g],1] <- alpha[state[i,g,steps[i,g]-1]] 
phi[i,g,steps[i,g],2] <- 1-phi[i,g,steps[i,g],1]
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])

#Final Substate
sub_phi[i,g,steps[i,g],1] <- sub_alpha[state[i,g,steps[i,g]]] 
sub_phi[i,g,steps[i,g],2] <- 1-sub_phi[i,g,steps[i,g],1]
sub_state[i,g,steps[i,g]] ~ dcat(sub_phi[i,g,steps[i,g],])

##  Measurement equation - irregular observations
# loops over regular time intervals (t)    

for(t in 2:steps[i,g]){

# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){ 
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]

#for each lat and long
#argos error
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec[argos_class[i,g,t,u],1:2,1:2])

#for each dive depth
#dive depth at time t
divedepth[i,g,t,u] ~ dnorm(depth_mu[state[i,g,t],sub_state[i,g,t]],depth_tau[state[i,g,t],sub_state[i,g,t]])T(0,)

#Assess Model Fit

#Fit dive discrepancy statistics
eval[i,g,t,u] ~ dnorm(depth_mu[state[i,g,t],sub_state[i,g,t]],depth_tau[state[i,g,t],sub_state[i,g,t]])T(0,)
E[i,g,t,u]<-pow((divedepth[i,g,t,u]-eval[i,g,t,u]),2)/(eval[i,g,t,u])

dive_new[i,g,t,u] ~ dnorm(depth_mu[state[i,g,t],sub_state[i,g,t]],depth_tau[state[i,g,t],sub_state[i,g,t]])T(0,)
Enew[i,g,t,u]<-pow((dive_new[i,g,t,u]-eval[i,g,t,u]),2)/(eval[i,g,t,u])

}
}
}
}

###Priors###

#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)

##Mean Angle
tmp[1] ~ dbeta(10, 10)
tmp[2] ~ dbeta(10, 10)

# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi

# prior for theta in 'foraging state'    
theta[2] <- (tmp[2] * pi * 2)

##Move persistance
# prior for gamma (autocorrelation parameter)
#from jonsen 2016

##Behavioral States

#gamma[1] ~ dbeta(5,2)      ## gamma for state 1
#dev ~ dunif(0.3,1)         ## a random deviate to ensure that gamma[1] > gamma[2]
#gamma[2] <- gamma[1] * dev

gamma[1] <- 0.7
gamma[2] <- 0.2

#Transition Intercepts
alpha[1] ~ dbeta(1,1)
alpha[2] ~ dbeta(1,1)

#Probability of behavior switching 
lambda[1] ~ dbeta(1,1)
lambda[2] <- 1 - lambda[1]

#Dive Priors
#Foraging dives are deepest
depth_mu[2,1] ~ dnorm(100,0.001)T(0,)
depth_sigma ~ dunif(0,50)
depth_tau[2,1] <- 1/pow(depth_sigma,2) 

#Resting Dives
depth_mu[2,2] ~ dnorm(20,0.01)T(0,)
depth_tau[2,2] <- 0.01

#Traveling Dives
depth_mu[1,1] <- 20
depth_tau[1,1] <- 0.01

#Dummy traveling substate
depth_mu[1,2]<-0
depth_tau[1,2]<-0.01

#Sub states
#Traveling has no substate
sub_alpha[1]<-1 

#ARS has two substates, foraging and resting
#Foraging probability
sub_alpha[2] ~ dbeta(1,1)

##Argos priors##
#longitudinal argos precision, from Jonsen 2005, 2016, represented as precision not sd

#by argos class
argos_sigma[1] <- 11.9016
argos_sigma[2] <- 10.2775
argos_sigma[3] <- 1.228984
argos_sigma[4] <- 2.162593
argos_sigma[5] <- 3.885832
argos_sigma[6] <- 0.0565539

#latitidunal argos precision, from Jonsen 2005, 2016
argos_alpha[1] <- 67.12537
argos_alpha[2] <- 14.73474
argos_alpha[3] <- 4.718973
argos_alpha[4] <- 0.3872023
argos_alpha[5] <- 3.836444
argos_alpha[6] <- 0.1081156

}"
,fill=TRUE)

sink()

##    user  system elapsed 
##   4.848   0.545 151.044

2.2 Chains

## # A tibble: 8 x 5
## # Groups:   parameter [?]
##   parameter Behavior   mean   upper  lower
##      <fctr>   <fctr>  <dbl>   <dbl>  <dbl>
## 1     alpha        1  0.423   0.674  0.185
## 2     alpha        2  0.073   0.125  0.031
## 3  depth_mu        1 10.000  20.000  0.000
## 4  depth_mu        2 66.638 102.165 31.965
## 5 depth_tau        1  0.010   0.010  0.010
## 6 depth_tau        2  0.005   0.010  0.000
## 7     gamma        1  0.700   0.700  0.700
## 8     gamma        2  0.200   0.200  0.200

3 Temporal Variation in Dive Behavior

## # A tibble: 4 x 3
## # Groups:   state [?]
##   state sub_state     n
##   <dbl>     <dbl> <int>
## 1     1         1    10
## 2     2         1    85
## 3     2         2     9
## 4     2        NA     1

3.1 Simulate dive depths

3.2 Diel

3.3 Month

4 Raw Argos Data

5 Comparison with 2d movement model

Movement model

##    user  system elapsed 
##   0.202   0.012  52.149

5.1 Chains

Where does the 3d predict foraging that the 2d misses?

Where does the 3d predict resting that the 2d misses?

Where does the 2d predict foraging that the 3d refines to traveling?

Where does the 2d predict foraging that the 3d refines to resting?

5.2 Posterior Checks

The goodness of fit is a measured as chi-squared. The expected value is compared to the observed value of the actual data. In addition, a replicate dataset is generated from the posterior predicted intensity. Better fitting models will have lower discrepancy values and be Better fitting models are smaller values and closer to the 1:1 line. A perfect model would be 0 discrepancy. This is unrealsitic given the stochasticity in the sampling processes. Rather, its better to focus on relative discrepancy. In addition, a model with 0 discrepancy would likely be seriously overfit and have little to no predictive power.

## # A tibble: 1 x 2
##   `mean(E)` `var(Enew)`
##       <dbl>       <dbl>
## 1  254.2023    96317.12

Foraging Model

Ben Weinstein

2017-11-30 20:49:23